INSTITUTE OF PHYSICS PUBLISHING
JOURNAL OF PHYSICS: CONDENSED MATTER
J. Phys.: Condens. Matter 16 (2004) 861–880
PII: S0953-8984(04)63457-6
Field emission theory for an enhanced surface
potential: a model for carbon field emitters
T C Choy, A H Harker and A M Stoneham
Department of Physics and Astronomy, University College London, Gower Street,
London WC1E 6BT, UK
E-mail: a.harker@ucl.ac.uk
Received 12 May 2003, in final form 23 December 2003
Published 30 January 2004
Online at stacks.iop.org/JPhysCM/16/861 (DOI: 10.1088/0953-8984/16/6/015)
Abstract
We propose a non-JWKB-based theory of electron field emission for carbon
field emitters in which, for electrons with energy in the vicinity of the order of ϑ
to the Fermi level, the effective (1/x) surface potential is strongly enhanced. The
model grossly violates the WKB validity criteria and necessitates an analytic
treatment of the one-dimensional Schrödinger equation, which we first obtain.
We determine ϑ (which is field-dependent) from the wavefunction matching
point close to the surface. For reasonable values of the surface parameters—
work function ϕ ≈ 2–5 eV, electron affinity χ ≈ 2ϕ and an empirical
electron loss factor σ ≈ 10−3 (and with no other adjustable parameters)—the
theory provides an intriguing agreement with experimental data from carbon
epoxy graphite composite (PFE) and certain graphitized carbon nanotube field
emitters. We speculate on the surface potential enhancement, which can be
interpreted as a massive (field-induced) dielectric effect of dynamic origin.
This can be related via time-dependent perturbation theory to second-order nonlinear polarizability enhancements at ultraviolet ∼3000 Å wavelengths near the
tunnelling region. Finally some exact mathematical results are included in the
appendix for future reference.
1. Introduction
The classic theories of electron field emission from a cold metal surface due to Fowler and
Nordheim (FN) [1] and later with image modifications due to Nordheim [2] were one of
the earliest applications of wave mechanics after its foundation in the early 1920s. FN
were originally motivated by their conviction of Sommerfeld’s (Fermi–Dirac-based) theory of
electrons in metals and thus their need to reject early interpretations of experiments by others
such as Millikan and Lauritsen [3], who suggested that the emitted electrons, ‘thermions’, are
to be distinguished from the conduction electrons in the metal1 . The Nordheim theory [2]
1 In the modern context, especially in non-metals for which tunnelling almost certainly comes from surface states
which are distinguished from the bulk, such ideas should probably not be scorned.
0953-8984/04/060861+20$30.00
© 2004 IOP Publishing Ltd
Printed in the UK
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T C Choy et al
actually consists of two parts: a full image potential, zero field (F = 0) but analytically
exact solution of the one-dimensional Schrödinger equation (based on modern day Whittaker
functions) and (for reasons of intractability at that time) an approximate image potential + finite
field (F = 0) theory using the Jeffreys’ method [4]. The latter is the now familiar modern-day
WKB approximation (henceforth we shall call this the JWKB approximation) [5–7]. This
approximate theory is also widely known as the Fowler–Nordheim (FN) theory by many later
workers, which is indeed a source of confusion. The FN–JWKB theory, with later corrections
for Nordheim’s erroneous reduction of elliptic integrals [8, 9], has become the landmark theory
for field emission studies from metal surfaces for the past 75 years, while the original exact
full image–zero field Nordheim theory has somehow disappeared into obscurity [10, 11].
With increased interest in field emission processes, especially for use in printable field
emitting (PFE) display screens [12], and the difficulty that the FN–JWKB theory is now
challenged in order-of-magnitude terms by the high current densities (of up to 0.1 A cm−2 )
observed in such systems, it is perhaps worthwhile re-examining some of its basic assumptions.
These are:
(1) Electrons are fermions and are governed by Fermi–Dirac statistics.
(2) They are thus characterized by well-defined quantities in the bulk, namely the Fermi
energy (E F at zero temperature T = 0 K) or chemical potential (µ at finite temperatures
T = 0 K), a work function ϕ and an electron affinity χ.2
(3) The effective surface potential, which is assumed to have a classical form outside the
surface:

for x < 0—region 1 (inside);
 −µ
(1)
V (x) =
e2
 ϕ − eF x −
for x > 0—region 2 (outside).
16πε0 x
Henceforth we shall take the zero of energy at the Fermi level, all energy being positive
above the Fermi level and negative below, with µ = (χ − ϕ). With minor exceptions,
such as the electric field F being in V µm−1 and the current density J being in A cm−2 ,
we shall use SI units and hence ε0 = 8.854 18 × 10−12 F m−1 is the permittivity of free
space and e = 1.602 18 × 10−19 C is the elementary charge.
(4) The JWKB approximation for the evaluation of the tunnelling probability T (W ), including
an integration over all energy levels W for the emission current with this probability.
In this paper we shall not critically examine each of these assumptions which would
also entail an examination of the criteria, regimes and conditions when each of them may
fail. Obviously, there are also regimes and conditions in which the theory has been rather
successful [9–11]. Our purpose is merely to innovate on reasonable grounds in a modern
context, perhaps incorporating necessary modifications, such as abandoning the JWKB
approximation as we shall see. Hence we shall state from the start that we shall not argue
against (1) and indeed retain a Fermi–Dirac-based supply function, see equations (6) and (43)
below. Our theory is based on a proposed modification to assumption (3), in which the 1/x
potential is significantly enhanced. The latter leads to a gross violation of the WKB validity
criteria (to be discussed later) and hence we are forced to reject assumption (4). Thus we are
compelled to consider an analytic solution of the one-dimensional Schrödinger equation close
to the surface. As well as exotic metal–vacuum interfaces, this theory may have further
applications for graphite–vacuum, graphite–insulator–vacuum, certain graphitized carbonnanotube–vacuum [13] and amorphous diamond-like carbon (DLC) Schottky junctions [14].
2
The temperature dependence and band-bending modifications of these parameters are more realistic considerations
which we will not pursue here.
Field emission theory for an enhanced surface potential
863
Since our approach can be generalized to other cases of interfaces between dissimilar media,
it may motivate a revisit of the systems discussed earlier by Mott and Gurney [15] and
Herring [16]. These other cases might include vacuum breakdown and electrical forming
processes [17]. In section 2, we shall discuss the basic features of our model, including a
rejection of the JWKB approximation in our case in view of the strong enhancement. In
section 3 we establish the analytic treatment of the one-dimensional Schrödinger equation
including its generalization to include the full image plus finite (F = 0) field problem, the
mathematical details of which we then postpone to the appendix B. While no exact closed
form analytical treatment is yet available, unlike the closed form Whittaker function analysis
historically first obtained by Nordheim [2] and later studied by MacColl [18], nevertheless we
have decided to include these results for future reference. Our solution is exact in the sense of an
infinite series in terms of Airy-type functions, even though for an enhanced potential (i.e. large
β̄, see later) the series is divergent. Nevertheless modern mathematical methods developed over
the last two decades of re-summation, such as Borel–Laplace transform asymptotic theory [19],
means that further progress towards a closed form solution is likely to be on the horizon.
Moreover we have found, as in our case, that in the limit of an infinitely large surface potential,
an exact solution is available in terms of Hankel functions of order one, which we shall be
using extensively. Hence re-summation of the exact finite field solution can be guided by this
limiting result, which could also be extended by perturbation analysis. Unfortunately we have
achieved no success in terms of a series expansion for the exact finite field problem in terms
of Whittaker functions, which is probably why this problem has not received much attention
for the last 75 years [10, 11].
In section 5 we are particularly interested in applying our theory to epoxy-graphite PFE and
graphitized carbon nanotube systems. We are intrigued not only by its removal of many ordersof-magnitude discrepancies between traditional FN–JWKB-based theories and experiment but
the better agreement at the lower onset fields in a traditional ln J versus 1/F plot. This was
achieved with few adjustable parameters (other than the work function ϕ, electron affinity
χ and a factor σ to account for empirical electron loss) for which the theory does not seem
to be very sensitive to, within reasonably acceptable values from experiments. This is to be
contrasted with the need for assumptions about large field enhancement factors ∼ 50–350
within current FN–JWKB-based theories. However, we do not dispute that field enhancement
factors have an important role in nano-structured systems. Indeed, for some (metallic based)
systems such as MoS2 nano-flowers, we have found quite good agreement between a classical
field-enhanced FN–JWKB theory and experiments [20], but this does not appear to be the
case for epoxy based carbon field emitters [12, 13, 21]. This suggests that the role of field
enhancement in such systems is more subtle and less straightforward, perhaps embedded via
dynamical polarization effects in a time-dependent tunnelling mechanism. In section 6 we
speculate on this new physics which our results seem to indicate. Here we will suggest an
interpretation of a massive (field-induced) dielectric effect at electromagnetic frequencies, of
the order of the inverse transit time across the surface, for electrons near the Fermi level.
Some open questions remain and suggestions for both further extensions of the theory and,
in particular, experiments to verify our observations will also be made. In the conclusion we
summarize our main results.
2. Preliminaries
As in the FN–JWKB theory, the role of the Fermi level, Fermi–Dirac statistics and, in particular,
the unique wave–particle duality of quantum mechanics are key concepts in our model. The
essentially equilibrium distribution of electrons is modelled by a one-dimensional (x-dependent
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T C Choy et al
only) Hamiltonian:
p2x
+ V (x),
(2)
2m
whose eigenvalue W is the x part of the energy measured with reference to the Fermi level [9].
At this stage some preliminary remarks about the potential V (x) (see equation (1)) are in order.
Our definition of the surface potential follows that of Good and Müller [9] and differs from
that of Guth and Mullin [22], who followed Nordheim [2] by connecting the outside classical
image potential to the bulk at some value x 0 such that
H=
e2
.
(3)
16πε0 x 0
We do not propose to do this, on the grounds that x 0 should be limited to no less than a Bohr
radius aB ∼ 0.5 Å. This would then artificially restrict the theory to systems with χ 6.8 eV, an
inconsistency dating back to the original paper of Nordheim, who subsequently quoted values
of χ (which is C in his notation) of between 10 and 20 eV [2]. For example, for tungsten, as
noted by Good and Müller [9], χ ≈ 10.3 eV and they must have found Nordheim’s proposal
troublesome. Nordheim’s, and later Guth and Mullin’s, inconsistency signals to us that on
the ångström scale the potential given by the classical expression equation (1) is an oversimplification, which modern techniques such as LDA/GGA have attempted to address [23].
However, we do propose later on that there is an x 0 ∼ 1 Å as the distance of closest approach
to the surface for the matching of the wavefunctions. There is no inconsistency here, though,
since length scales less than x 0 ∼ 1 Å are outside the regime of physics considered in our
model. We propose that the behaviour of the potential near x 0 is highly significant, which is
in contrast to Good and Müller [9] whose ‘potential is not expected to have a significance at
the point x = 0’. This was earlier also noted by Guth and Mullin [22], but they did not pursue
the matter further.
Next the current density J of the emitted electrons is usually given by an integral over the
energy W :
∞
P(W ) dW,
(4)
J =e
χ =µ+ϕ =
−∞
where P(W ) dW gives the number of electrons within dW that emerge from the surface per
second per unit area. However, it is a fundamental assumption in our model that we shall, in
fact, replace equation (4) by
0
P(W ) dW,
(5)
J =e
−ϑ
as the dominant term, since all other contributions are many orders of magnitude less. As we
shall see later, this includes the finite-temperature T = 0 K corrections to our theory, which
are down by at least five orders of magnitude at room temperature, for the field strengths we
consider. The quantity P(W ) is usually expressed as the product of a quantum mechanical
tunnelling transmission coefficient T (W ) and a Fermi-type supply function N(W ) so that
4πmkB T
W
,
(6)
N(W ) =
ln 1 + exp −
h3
kB T
in which kB is the Boltzmann constant. We have no doubt that a substantial modification of
the JWKB tunnelling probability:
x2 1/2 8m
TJWKB (W ) = exp −
(V
(x)
−
W
)
dx ,
(7)
h̄ 2
x1
Field emission theory for an enhanced surface potential
865
in which x 1 and x 2 are the classical turning points of the potential V (x), is a prerequisite for
any viable theory for the carbon emitters [21]. There are at least two reasons for this.
The first reason is that a reduction of the tunnelling barrier height is inevitably necessary
for any significant emission current. In the classical theory this is the Schottky image reduction
given by
√
eF
φ = e
= 3.7947 × 10−2 eF eV1/2 µm1/2 ,
(8)
4πε0
which for a work function ϕ would require fields of the order of
Fb = 694ϕ 2 eV−2 V µm−1 .
(9)
In our theory we shall mimic this barrier reduction in a different way, see section 6 later. For
now, let us examine the classic FN–JWKB formula [9] for the emission current density:
σ a F2
exp[−bυ(yF )ϕ 3/2 /F],
t (yF )2 ϕ
a = 1.5414 × 102 A cm−2 eV µm2 V−2 ,
JFN =
b = 6.8309 × 10 eV
3
yF = 3.7947 × 10
−2
−3/2
F
1/2
(10)
−1
V µm ,
/ϕ eV µm1/2 V−1/2 ;
in which t (yF ) and υ(yF ) are given in terms of elliptic integrals [8, 9] and all fields F are
measured in V µm−1 in this paper. We have included a factor σ to account for empirical
loss factors, such as those due to cathode–anode geometrical factors, back-contact resistance
and imperfect supply, since for composite and granular systems we expect a larger fraction of
electron loss than for a clean metal surface in general. Equation (10) shows that for a typical
work function ϕ ∼ 2 eV, and applied fields in the range 1–16 V µm−1 , a field enhancement
factor F → F with of up to 50 is necessary to produce currents of the order found for carbon
emitters [21] and up to ∼ 350 for carbon nanotubes [13], see later. More importantly, in view
of the weak dependence of the elliptic-function-based υ(yF ), the behaviour of ln J remains
essentially that of 1/F which is not in accord with data for carbon field emitters [13, 21].
However, we must emphasize that there are systems such as MoS2 nano-flowers [20] that are
in accord with the classical field-enhanced FN–JWKB description. Our concern in this paper
is with systems such as epoxy graphite and certain graphitized nanotube carbon field emitters,
which appear to be in disagreement with FN–JWKB theory (see later).
The second reason is the WKB criteria. It is a well known result that the validity of
the WKB approximation involves a criterion, that can be derived from the requirement that
the variation of the deBroglie wavelength of the electron must be less than the dimension of
the region of tunnelling [24]. This restriction on the validity of the FN–JWKB theory is well
known to workers in the field, for example in the treatment of thermionic emission by Guth and
Mullin [22] who attempted to improve the FN–JWKB theory by an exact analytic treatment
of a parabolic approximation to the top of the barrier. Our concern is, however, with regions
close to the surface of the order of ångströms to which tunnelling presumably occurs. Then
the WKB validity criteria can be written as [24]
∂V p3 = h̄ 3 k 3 ,
(11)
mh̄ F
F
∂x which, for fields 1 < F < 20 V µm−1 , is dominated close to the surface by the image term,
so that the above can be rewritten as
e2
x 0 1890.19
eV Å2 .
(12)
16πε0 x 0
λF λ2F
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T C Choy et al
For x 0 ∼ 1 Å and λF ∼ 7 Å the ratio ζ (of the LHS to the RHS) for the inequality in equation (12)
is about 0.6, as also noted previously by Guth and Mullin [22] for the case of tungsten but
using a different analysis. The situation is worse in our case when we shall later propose a
strongly enhanced surface potential by at least two orders of magnitude. Unfortunately we
are not aware of any detailed analysis for the degradation in the JWKB approximation as a
function of the ratio ζ , nor of any systematic means for improving it when ζ 1, although
some recent studies are beginning to emerge in the mathematical literature, due to advances in
resurgent analysis [19].
The reader should note that our previous remarks are not meant to be a criticism of the
JWKB approximation nor the FN–JWKB theory in general (see also remarks in the third
paragraph of the last section). We merely need to stress that in the region close to the
surface, and especially when the surface potential is strong, then there are reasons not to
invoke a simplistic JWKB approximation. In fact, as pointed out by Guth and Mullin [22], a
more sophisticated treatment of the FN–JWKB is possible. In their model they made use of
connection formulae derived via an exact analytic solution of the parabolically approximated
barrier top potential (which unfortunately involves rather complicated parabolic cylinder
functions). Using the exact (F = 0) Whittaker function solution of Nordheim [2] and later
MacColl [18] as a test case, they claim to have found transmission coefficients that are in
agreement numerically. We do not know to what extent this agreement still holds for large
fields and in particular what other modifications are necessary for strong image potentials
(ζ 1). There is also a puzzling point in that the (F = 0) case (being of 1/r only) does not
really have a parabolic barrier top and the agreement could therefore perhaps be fortuitous.
This subject is no doubt worth further studies in view of our exact analytic solution in the form
given in appendix B.
3. Imageless triangular barrier
We shall review the image-free tunnelling problem given by the following Schrödinger equation
which is important for orientation:
2m
(13)
ψ (x) + 2 [W − V0 (x)]ψ(x) = 0,
h̄
where again W is the energy eigenvalue of the Hamiltonian H but now with the potential
V0 (x), which does not contain the image term:
−µ
for x < 0—region 1;
V0 (x) =
(14)
ϕ − eF x
for x > 0—region 2.
This is a well known problem to theoretical workers in the field dating back to Fowler and
Nordheim (FN) [1] and also recent review articles [10, 11], including more recently Jensen
et al, who used these exact results as a basis for analytical approximations for the image
rounded barrier problem, which they claim to have been validated by numerical studies [25].
For later convenience we shall recast equation (13) into the more compact form
ψ (x) + ( + αx)ψ(x) = 0,
(15)
2meF
= cα eF,
h̄ 2
2m(W − ϕ)
= cα (W − ϕ);
=
h̄ 2
cα = 2.6246 × 107 eV−1 µm−2
(16)
with
α=
Field emission theory for an enhanced surface potential
867
in which F is measured in V µm−1 as usual. To the best of the authors’ knowledge this
graduate-level quantum exercise does not seem to have found its way into many textbooks,
although variants of it have been scattered around [27–30]: Modinos [26] simply cites Gadzuk
and Plummer [10]. Following Landau and Lifshitz [31], we shall transform to the new
variable [26] ξ = (x + /α)α 1/3 . Then equation (15) in region 2 becomes the typical Airy
equation3:
ψ (ξ ) + ξ ψ(ξ ) = 0.
(17)
However, some care needs to be exercised in selecting the appropriate type of Airy function,
since the standard Ai function has asymptotic form [31, 32]:
Ai(−ξ ) ≈ ξ −1/4 sin( 23 ξ 3/2 + π/4).
(18)
As this also contains a reflected wave, it is not appropriate to our boundary conditions. The
correct approach is to revert to the original 1/3 fractional-order Bessel function definition for
the Airy functions and then construct appropriate linear combinations to yield the forward
travelling wave solution only. This approach, of course, leads to an appropriate modified
Hankel function which, as we shall see, has the correct asymptotic form required. Other
combinations of Bessel functions are possible: for these the reader may refer to the literature
and texts [10, 11, 27, 29, 30]. With this proviso, we can write down the solution for the
wavefunctions of equation (17) as
ikx
e + r e−ikx
in region 1;
(19)
ψ(x) =
−iπ/3
t[u 1 (ξ ) − e
u 2 (ξ )]
in region 2,
where k = (2m(W + µ)/h̄ 2 )1/2 and the u 1 and u 2 are given by Bessel functions of order 1/3,
here defined as
u 1 (ξ ) = 13 (πξ )1/2 J−1/3 ( 23 ξ 3/2 ),
(20)
u 2 (ξ ) = 13 (πξ )1/2 J1/3 ( 23 ξ 3/2 ),
(21)
and
respectively. Note, however, that analytical continuation to negative ξ requires some care: see
appendix A. It is then straightforward to obtain the reflection and transmission amplitudes by
matching the wavefunctions and their derivatives at the interface x = 0. We provide the results
here in terms of H1/3 = u 1 − e−iπ/3 u 2 , which is a slightly modified Hankel function:
2
t=
1/3 H1/3 (ξ0 ) − iλ0 H1/3
(ξ0 )
(22)
1/3 H1/3 (ξ0 ) + iλ0 H1/3 (ξ0 )
r=
,
1/3 H1/3 (ξ0 ) − iλ0 H1/3
(ξ0 )
1/3
where ξ0 = ξ(x = 0) = ¯ = /α 2/3 here, while λ0 = α 1/3 /k. An important check must be
made to these formulae which requires that the transmission coefficient:
1/3
λ0
|t|2 ,
(23)
4
and reflection coefficient, R = |r |2 , satisfy the unitarity property T + R = 1. This is easily
checked to be the case when appropriate use is made of the Wronskian identity:
1
(24)
u 1 u 2 − u 2 u 1 = √ .
2 3
T =
3
The advantage of using the Airy function formulation is that it naturally takes care of the long range boundary
condition posed by the electric field.
868
T C Choy et al
For sufficiently small fields, we can obtain an asymptotic expansion (assuming ξ0 →
∞ but λ0 → 0) that yields
1/3
Tasymp (W ) ≈
4λ0 |ξ0 |1/2
2/3
(1 + λ0 |ξ0 |)
e− 3 |ξ0 |
4
3/2
4[|(W − ϕ)|(W + µ)]1/2 − 4 |ξ0 |3/2
e 3
,
(25)
µ+ϕ
the pre-factor being a result not obtainable within ordinary WKB. The reader may like to
note that this result is identical with equation (18) of FN [1], who also gave a more accurate
asymptotic formula in their earlier equation (17) FN [1], and more recently a higher-order
asymptotic expansion of this pre-factor has been given by Jensen [11]. For the sake of
illustration, let us set µ = ϕ for the moment. Then we see that the above pre-factor for
W = 0, i.e. Tasym (0), differs from the WKB result by a mere factor of two, i.e.
=
4 3/2
Tasymp (0) ≈ 2e− 3 ξ0 .
(26)
That this asymptotic result could be problematic as we approach the top of the barrier can
be seen from the fact that if we naively allow the fields to become arbitrarily large and adopt
equation (26), then we would arrive at an unphysical result in which T > 1, which is absurd.
No doubt this can only occur for very large fields F ∼ 104 ϕ 3/2 V µm−1 in general for the
imageless case, but if a significant barrier reduction mechanism operates (which effectively
reduces ϕ even to millielectronvolts), then it would be necessary to use the original formula
equations (22) and (23). These observations are not new and are well known to workers in
semiconductor devices [27, 33] but are worth reiterating in any case.
Once again we emphasize that our purpose is not to criticize the WKB approach in general,
but to merely note that its prediction of large transmission coefficients at high applied fields,
without knowledge of the pre-factor, can be problematic. Unfortunately this caution does not
seem to have been emphasized by earlier workers such as Good and Müller [9] nor by Murphy
and Good Jr [34]. The latter have derived widely used criteria [35] for the validity of FN–
JWKB theory on the basis of the WKB integrand properties equation (7) alone, which may be
inadequate. Indeed any sophisticated beyond-WKB procedure, whose purpose it is to recover
the correct pre-factor, as in equation (26), may not be an altogether worthwhile exercise in
light of the above remarks. The reader should note in particular the well known result from
elementary quantum mechanics that the transmission coefficient is, in general, not unity at the
top of the square barrier [36].
4. The full image rounded triangular barrier
We now turn to the quantum tunnelling problem including the effect of the image term as in the
original Nordheim [2, 18] theory. As far as the authors are aware, there have not been many
serious attempts in the past at seeking an exact analytic solution of the full one-dimensional
Schrödinger equation. Our purpose here is twofold. Firstly, we shall document our attempts
and comment on the successes and failures. This work has led us to interesting territory
and also educated us in some important advances in modern mathematical analysis, not widely
known to standard mathematical physics [19, 37] and differential equations [38, 39]. Secondly,
and more importantly, our analysis here has motivated us to innovate on a choice of surface
potential that will be applied to carbon emitters in the next section.
The full image + field Schrödinger equation (15) now becomes
β
ψ(x) = 0,
(27)
ψ (x) + + αx +
x
Field emission theory for an enhanced surface potential
869
where the image potential is characterized by the parameter β = me2 /(8πε0 h̄ 2 ), but in view of
our lack of knowledge of surface potentials, we might perhaps consider β as a free parameter,
see later. Once again, transforming to the parameter ξ as before, we now have the equation
β̄
ψ (ξ ) + ξ +
ψ(ξ ) = 0,
(28)
ξ − ¯
where β̄ = βα −1/3 and ¯ = α −2/3 . Clearly the image-free case β̄ = 0 reduces to the usual
Airy differential equation (17). Equation (28) can be rewritten as
(ξ − ¯ )ψ (ξ ) + [ξ(ξ − ¯ ) + β̄]ψ(ξ ) = 0.
We shall in fact further rewrite this as
(ξ − λ1 )(ξ − λ2 )
ψ (ξ ) +
ψ(ξ ) = 0,
(ξ − ¯ )
in which the two roots of the quadratic are given by
1
1
2
and
λ2 = 2 ¯ − ¯ 2 − 4β̄ .
λ1 = 2 ¯ + ¯ − 4β̄
(29)
(30)
(31)
Let us first discuss the WKB approximation [4–7, 24], which is obtained by approximating
the wavefunction as
ξ
iπ
1
p(ξ ) dξ +
ψ(ξ ) = √
exp i
,
(32)
4
p(ξ )
where
(ξ − λ1 )(ξ − λ2 ) 1/2
;
(33)
p(ξ ) =
(ξ − ¯ )
and the lower limit in the integral in equation (32) is the smaller of λ1 or λ2 , which depends
on the sign of ,
¯ i.e. above or below the barrier, respectively. The evaluation of the integral
in equation (32) is then an exercise in elliptic integral reduction which we need not repeat
here [2, 9, 34], which fortunately nowadays is greatly aided by tools such as Mathematica v4.1.
Equation (30) itself does not appear to have a closed form solution in terms of known functions,
although several series methods of solution are available, depending on the parameters; see
appendix B. We may as well write down the exact closed form solution for the case (F = 0)
in terms of the appropriate Whittaker function, due to Nordheim [2], as a start. This can be
written in terms of the scaled parameters z = 2¯ 1/2 y, where y = ξ − ¯ (which will be important
to us later):
β̄
;
(34)
2¯ 1/2
where the Whittaker functions Wλ,µ (z) are known to satisfy the differential equation [40]
1
− µ2
1 γ
d2 W
4
+ − + +
W = 0.
(35)
d2 z
4 z
z2
In fact, in terms of the variable z the differential equation we seek (equation (27) or (28)) is
less complicated looking than equation (35), being
1
1 γ
ψ (z) +
+ + λ̄z ψ(z) = 0,
λ̄ = 3/2 ;
(36)
4 z
8¯
and it is a puzzle to us that it has received so little attention. However, our attempts at
a series solution based on the Whittaker functions have not met with success, in view of
the incompatibility of the recursion and orthogonality properties of these functions with the
ψ(z) = Wiγ , 12 (iz),
γ =
870
T C Choy et al
Figure 1. Plot of the classical surface image potential for a field enhancement factor of = 50.
Curves are drawn in decreasing order for F = 16 (full), 10 (long dash), 5 (short dash) and 1 (dotted),
respectively, in units of V µm −1 . V (x) is in electronvolts, with the work function ϕ = 2.0 eV
and the distance x from the surface is in ångströms. Note the Schottky barrier is reduced by about
53.6% for the largest field.
linear field term in equation (36). This state of affairs is puzzling which is in contrast with
the analogous problem of the hydrogen atom in an electric field, already noted by the early
pioneers (see references in FN [1]). The latter can be treated exactly by a rapidly convergent
infinite series expansion in terms of Laguerre/confluent hypergeometric functions as in the
Stark effect/van der Waals molecule problem, see, for example, [41]. We do not know if this
merely reflects our ignorance of the properties of Whittaker functions or otherwise, for in the
appendix B we shall show that other series methods of solution exist. We will therefore not
discuss these results further here except to say that they are, in general, series expansions in
the parameter β̄/¯ . Unfortunately for large fields or enhanced β̄, which are the main interest
to us here, these series are not useful without some sophisticated re-summation procedure.
Fortunately for us this is not necessary for the purpose we have at hand, which is to apply
the enhanced potential theory to a model for carbon field emitters, for which an alternative
(perturbation) approach is more suitable.
5. Model for carbon field emitters
In recent years there has been widespread excitement in carbon based field emitters [12, 13, 21]
which have the potential, (already realized) for cost-effective large area display systems. The
motivation for our work is based on the observed high current density and, as we shall see, the
non-exponential behaviour of these emitters. Our first observation is the relative magnitudes
of the various coefficients, especially in the form equation (36). In particular we shall focus
on the barrier top (see figure 1) which occurs when β̄ = ¯ 2 /4 and hence also when γ = ¯ 3/2 /8
becomes large. We note that this can be achieved through a large field enhancement effect of the
order of Fb /F in FN–JWKB theory, see equation (9) as well as from yF = 1 in equation (10).
However, the current will be too large in general (since JWKB predicts T = 1) and the field
dependence of ln J will remain that of 1/F, which falls too steeply as we shall see later. This is
a consequence of the fact that, in the classic FN–JWKB theory, transport is mainly via quantum
tunnelling through the classically forbidden region, even if the potential has been significantly
lowered (see figure 1).
We will not discuss the mechanism by which the barrier lowering actually occurs in
our model. Instead we shall adopt it as a working hypothesis, but we do not expect that all
Field emission theory for an enhanced surface potential
871
Figure 2. Plot of our effective surface potential as in figure 1. Curves are drawn in decreasing
order for F = 16 (full), 10 (long dash), 5 (short dash) and 1 (dotted) V µm −1 , respectively. V (x)
is in electronvolts, with the work function ϕ = 2.0 eV and the distance x from the surface is in
ångströms. Note that there is no classical Schottky barrier here and the magnitude of V (x) suggests
that this potential can only be of very short duration, of the order of femtoseconds (see the text).
electrons in the band shall enjoy this opportunity, except for a narrow strip from W = 0 to
ϑ very close to the Fermi level. Indeed we do not even expect that this (enhanced potential)
barrier lowering phenomenon takes place in a general static sense. It is probably dynamic
and needs to occur only over a timescale of the order of the transit time, which is about a
few femtoseconds (10−15 s). Previous workers, such as Guth and Mullin [22], have noted the
dominance of the image term for small distances of the order of ångströms from the surface.
Our hypothesis here is to allow for a large enhancement in β, and hence β̄, that effectively wipes
out the barrier during the transit across the surface layer for a special group of near-Fermi-level
electrons, which we shall call the ϑ electrons. We shall discuss the physical interpretation of
this assumption later. For now we shall merely state our key hypothesis that, for the purpose
of calculating tunnelling probabilities for the ϑ electrons, the parameter β̄ shall be replaced
by β̄ = ¯ 2 /4 or equivalently γ = ¯ 3/2 /8 in equation (36). In view of the magnitude of the
parameter ¯ it then suffices to obtain a solution in the limit in which this parameter is large
and then treat the rest of the terms in the Schrödinger equation, such as in equation (36), as
perturbations. This is the sole mathematical basis of our theory, as well as its main underlying
physical assumption. For ϕ = 2 eV and 1 < F < 16 V µm−1 , then ¯ ranges from about 594
to 94, which should indeed justify treating the neglected terms as perturbations. In fact, a plot
of the now enhanced potential (see figure 2) shows that the neglected terms are insignificant
for the largest field F = 16 V µm−1 considered, up to some 1000 Å. This figure further shows
that such a potential cannot be interpreted in the usual static sense, but we shall defer these
discussions to section 6.
We expect that, over the distance of a few nanometres, tunnelling would have occurred from
the surface if it is possible at all. Thus the long range departures of our approximate effective
potential from the full potential will only contribute in a perturbative way. In particular, for
the matching conditions we are most interested in the solution near to the origin x = 0, which
corresponds to ξ(0) = ξ(x = 0) = ,
¯ so that we can go back to the variable y = ξ − ¯ , which
is small for our purpose y ¯ and then equation (28) can be well approximated by
yψ (y) + (y + ¯ /2)2 ψ(y) ≈ yψ (y) + (¯ /2)2 ψ(y) = 0.
(37)
The appropriate solution for the above is now given by the Hankel function of order 1 which,
872
T C Choy et al
upon transforming back to the original variables ξ , can be written as
π
¯ (ξ − ¯ )1/2 H1(1)(¯ (ξ − )
¯ 1/2 ).
(38)
ψ(ξ ) = eiπ/4
2
We can now obtain the transmission coefficient as was done in the previous section by matching
the wavefunction and its derivative, but now at ξ0 = ξ(x = x 0 ), whose value we shall determine
later. The result is then given by
1/3
T =
2¯ 2 λ0
1/3
2/3
|ψ(ξ0 )|2 + ¯ 2 λ0 + λ0 |ψ (ξ0 )|2
,
(39)
where the derivative of ψ is easily shown to be given in terms of the Hankel function of order
0:
π 2 (1)
¯ H0 (¯ (ξ − ¯ )1/2 ).
ψ (ξ ) = eiπ/4
(40)
8
As usual we can verify that this solution satisfies unitarity T + R = 1 through the use
of the appropriate Wronskian identities for the Hankel functions. That this is satisfied
exactly in equation (37) for the modified potential is important and future perturbative studies
should respect this unitarity condition. In fact, we are not aware if other approximate
procedures [22, 25] are so respectful of the unitarity condition. We believe this is necessary
for any physically self-consistent theory for large transmission. We should remind the reader
that the WKB approximation violates unitarity in an exponentially small way, provided the
transmission coefficient is exponentially small, see section 50 [24]. Equation (39) has a weak
logarithmic behaviour (due to the 1/x potential near the origin) in ψ . However, the physics
of our model does not go below the order of 1 Å. We shall therefore invoke this by letting
x 0 = 1 Å as the matching point on the surface. Thus we may write to the leading order in
large quantities
4π
W +µ
,
(41)
T ≈
3/2
1/2
3/2
2
|¯ | [ln(δ |¯ | )] |W − ϕ|
where δ(x 0 ), which we emphasize is a function of the matching point x 0 , will be determined
below. Physically it is reasonable that we may choose to connect the wavefunctions on the left
to the right at the point x 0 , since our theory cannot describe sub ångström processes, which
must involve deBroglie wavelengths λ λF . The equality of δ(x 0 ) so chosen with the energy
of the ϑ electrons, i.e. ϑ ≈ δ(x 0 )ϕ, is an additional hypothesis, which seems an obvious
choice. However, its justification lies outside the realm of the present theory, for which we can
only speculate at this stage, see section 6. Equation (41) is the key result of our theory, which
differs from the JWKB approximation (T = 1) at the top of the barrier. As we have taken
x 0 as about two Bohr radii, i.e. x 0 ≈ 2aB = 1 Å,4 then by definition we can now determine
ξ0 = ξ(x = x 0 ) from ξ0 − ¯ = δ ¯ = α 1/3 x 0 and thus we have
eF x 0
10−4 eF
ϑ
=
=
µm.
(42)
ϕ
ϕ
ϕ
Using equation (41) we can now estimate the current density near the barrier top. We shall
do this at finite temperatures using equations (5) and (6). The former can now be written as
ϕω
kB T
meϕ 2 0
ln 1 + exp −
dω,
(43)
T (ϕω)
J=
ϕ
kB T
2π 2 h̄ 3 −δ
δ = δ(x 0 ) =
4 By way of comparison, the typical tunnelling length in FN–JWKB theory (without field enhancement) is of the
ϕ
order of = eF
∼ 104 Å to 103 Å for 1 < F < 16 V µm−1 , with ϕ = 3 eV.
Field emission theory for an enhanced surface potential
873
where we have introduced a dimensionless integration variable: ω = W/ϕ. Note that due to
the assumptions (1) and (2) in section 1, with all electrons belonging to the same band, the
parameters δ, and hence also ϑ, do not enter the derivation of the supply function N(W ) in
equation (6).
Now in view of the smallness of δ, and thus the exponent of the exponential throughout
the region of integration, the above can be rewritten as
meϕ 2 0
kB T
1
J≈
T
(ϕω)
ln
2
−
ω
dω.
(44)
ϕ
2
2π 2 h̄ 3 −δ
An estimate of each term can now be made from equation (42) which shows that the finite
temperature term would go as
δkB T
kB T (eF) × 10−4
∼
,
(45)
ϕ
ϕ2
which is at least five orders of magnitude down from the leading term for the temperature and
fields we consider. We have to remark that the temperature dependence of δ, which we should
emphasize as δ(x 0 , T ), is outside the realm of the present theory. However, we believe it is
unlikely that the already small value of δ can be sustained to very high temperatures of the
order of ϕ/kB . Hence any serious discussions of the high temperature dependence are probably
premature at this stage.
In view of the weak dependence of the logarithmic term in equation (41) the integration
can proceed by extracting the leading field dependence in its argument. The final result we
obtain is given by
µ
σ cA ϕ −3/2
(eF)3 A cm−2 ,
(46)
JCH ≈
1/2
2
[ln(c B ϕ/(eF) )]
ϕ
where the constants are cA = 9.924 × 10−2 eV−3/2 µm3 and cB = 51.231 eV−1/2 µm−1/2 ,
respectively, and σ is the same empirical electron loss factor as introduced in the FN–JWKB
formula equation (10) earlier.
Note that the field dependence of the current is in this case F 3 which differs from the
FN–JWKB prefactor which goes as F 2 and there is no exponential dependence. We can
now compare equation (46) with the standard FN–JWKB formula equation (10) and with
experimental data [21]. To do this we need to take into account some empirical loss factors.
This must include the cathode–anode probe geometry since ours is a one-dimensional model.
Even if the cathode emitting surface is kept very small, comparable to the probe tip (∼1 mm)
in most cases, we do not expect all electrons to be collected. The imperfect supply is a further
problem for composite granular systems, since the Fermi-supply function is an ideal case: some
electrons could be trapped or recombined with holes at various sites, especially near contacts.
These studies require more detailed knowledge of the experimental conditions. However, we
expect that they do not vary significantly in order-of-magnitude terms with the field over the
range considered here. Hence for simplicity we have adopted a consistent approach for both
the FN–JWKB and our model by including a factor σ in the two formulae. For the FN–JWKB
formula we further include a field enhancement factor , i.e. F → F, see figure 3. We
found that we need to have ∼ 50 to get the order of magnitudes shown for the FN current
density. An interesting observation is that the larger departures at the higher fields in our case
may be due to the neglected terms in equation (36), that is the 1/4 factor and the λ̄, which
a future perturbative calculation may be able to improve. We emphasize that we have only
two adjustable parameters, namely ϕ and µ, to which the results are not too sensitive, within
the reasonable values expected. The parameter σ is set by the FN–JWKB current so that we
are comparing like with like. For FN–JWKB, there is an additional parameter , which is the
874
T C Choy et al
Figure 3. Logarithmic log10 J versus 1/F plot of the emission current in our model for epoxygraphite PFE material. The black dots are experimental points from [21]. The full curve is our
result which is given in equation (46). The dotted curve is the same but does not include the
logarithmic field factor in the denominator. The broken straight line is the FN–JWKB result JFN ,
as given in equation (10) with a field enhancement factor of = 50. J is in units of A cm −2 and F
is in units of V µm −1 . The work function ϕ = 2.0 eV, with µ = 2ϕ and σ = 10−3 in both cases.
Figure 4. Logarithmic log10 J versus 1/F plot of the emission current in our model for graphitized
carbon nanotube samples on an AAO template. The black dots are experimental points from [13].
The full curve is our result which is given in equation (46). The broken straight line is the FN–
JWKB result JFN as given in equation (10) with a field enhancement factor of = 350. J is in
units of A cm −2 and F is in units of V µm −1 . The work function ϕ = 5.0 eV, with µ = 2ϕ and
σ = 4.91 × 10−3 in both cases.
enhancement factor, and the FN–JWKB current density is very sensitive, in order-of-magnitude
terms, to its value. Especially interesting is a comparison with data from graphitized carbon
nanotubes, see figure 4. These intriguing results inspire us to seek an interpretation for the
strong potential enhancement, which we shall now discuss.
6. Discussions
In this section we will make no pretense that we are mainly speculating. The two key ingredients
in the theory require some physical interpretation. These are (i) the strong surface potential
enhancement and (ii) the connection between the matching point x 0 and the ϑ electrons.
We shall first address the potential enhancement factor by noting that our barrier lowering
assumption β̄ = ¯ 2 /4 in the last section can be rewritten in terms of a suppression of the
Field emission theory for an enhanced surface potential
875
effective dielectric constant κs for the surface potential, see equation (27). This can be deduced
from
β
,
(47)
φ(r ) = −
κs r
whose classical image theory value for κs is given in terms of the bulk material static dielectric
constant κb as
κs =
κb + 1
.
κb − 1
(48)
The well known value for κb is 11.9 for the case of silicon, making κs ≈ 1.18. The factor of
dielectric suppression in our model potential is therefore of the order of (from β̄ = ¯ 2 /4)
κs =
ϕ2
e3 F
F
F
=
≈
,
2
2
(W − ϕ)
Fb (W − ϕ)
Fb
(49)
for the ϑ electrons. This massive dielectric suppression is therefore of the same order as
the field enhancement factors required in classical field-enhanced FN–JWKB theory to obtain
the same magnitude of current density. Such a suppression cannot occur in a static theory,
since static dielectric constants must be greater than unity on thermodynamic grounds [42].
Thus we now need to consider a dynamical picture, whose key parameters may be the Fermi
wavelength λF and a characteristic angular frequency ωF , which will be typically of the order
of the inverse transit time τF = λF /vF ∼ 10−15 s, as mentioned before. It is well known that
for the case of silicon (see, for example, [33] section 5.3), the electron transit time τF ∼ 10−14 s
is slow by comparison with the dielectric relaxation time, which is controlled by the plasmon
frequency τp ∼ 10−16 s. Hence the static polarizability prevails and the classical Schottky
mechanism holds for silicon. Since graphite is known to have a low density of states near the
Fermi level, we would expect a significant modification of this picture. We note in passing
that the transit time is still a subject of some contention [43]. Indeed a proper time-dependent
theory of quantum tunnelling that goes beyond WKB is still not available at present, to the best
of our knowledge. Our work here reveals that this development is a matter of some urgency.
Therefore we expect our estimate could be out by orders of magnitude. Nevertheless if we
accept this femtosecond estimate, then the electromagnetic wavelengths concerned would be
of the order of 3000 Å which is in the near-ultraviolet range. This effect is likely to be confined
only to the order of λF from the surface, since if not, there would be consequences more easily
detectable than in field emission. An examination of equation (49) further shows that the effect
is non-linear. A polarizability that is linear in F implies a quadratic order dipole polarization
from a time-dependent perturbation theory perspective [44]. Hence optical measurements that
probe the surface, perhaps using evanescent waves for example, might be extremely revealing.
We do not know to what extent this property is related to the graphitization of the carbon
particle surfaces, as details have not yet been systematically investigated to the best of our
knowledge.
Next we wish to state clearly that we do not yet know the underlying physics, at this
stage, for relating the energy of the ϑ electrons through the closest approach matching point
x 0 as defined by δ(x 0 ) in equation (42). We adopted this ansatz as the most obvious choice.
It is intriguing, however, that the latter is merely the ratio between an atomic polarization
energy ϑ = eF x 0 versus the work function ϕ, see equation (42). Heuristically this implies
that one can expect the dielectric suppression, which must only involve a fraction δ of surface
atoms, would also involve the same fraction of electrons near the Fermi level in any dynamical
dielectric processes. However, one has to note that this choice may well be peculiar only
to the 1/x potential, which could also be modified at distances of the order of the atomic
876
T C Choy et al
size. Indeed a systematic study using a more sophisticated enhanced potential, which includes
dipole, quadrupole and higher-order multipole terms, would be very interesting, since this
would lead to angular predictions of the emission current density that might be detectable
experimentally. The relevant parameters and their connection with the ϑ electrons will remain
a challenge for a more sophisticated dynamical theory. The order of the onset field for the
proposed dielectric phenomenon to occur, which is F ∼ 1 V µm−1 , is particularly important
to further elucidate and must not be forgotten.
7. Conclusion
In conclusion, we have presented a relatively high current field emission model which is based
on a strongly enhanced surface potential. The enhancement leads to a significant barrier
lowering that mimics the classical field-enhanced Schottky barrier lowering mechanism. In
view of this strong enhancement, which grossly violates the WKB validity criteria near the
surface, we are compelled to pursue an analytic solution of the one-dimensional Schrödinger
equation. Fortunately we were able to do this, with less effort than the classical image (zerofield) Nordheim/MacColl solution [2, 18]. By treating the smaller terms as perturbations and
the assumption that only a thin layer of electrons near the Fermi level (which we call the ϑ
electrons) are involved in the process, we found an intriguing agreement with experimental
data. This includes epoxy graphite composite PFE [21] and graphitized carbon nanotubes on
an AAO template [13] field emitters, both in terms of the order of magnitude as well as the
shape of the curve in a ln J versus 1/F plot, particularly at lower onset fields. We further
interpret this result as a massive dielectric suppression involving only the near-Fermi-level
ϑ electrons and the surface atoms. Further work is necessary, which should include more
accurate calculations, perhaps employing the results from the exact infinite series solutions
given in the appendix. Understanding the microscopic physics of this phenomenon might
involve dynamical effective medium concepts [45] and also shape enhancement factors [46].
We may perhaps further speculate that, in a microscopic theory, our effective potential, see
figure 2, is only meaningful as a relevant Fourier component of a time-dependent fluctuating
potential. Hence a proper time-dependent tunnelling theory that is coupled to an underlying
mechanism for the dielectric suppression involving the ϑ electrons will be a highly worthwhile
challenge.
Postscript: after this work was completed, we have found that our solution equation (38)
has also been previously obtained by MacColl [18] in the context of zero glancing angle
reflection from metals with no applied fields. His reflection coefficient R is given by R = 1− T
in terms of our T in equation (39).
Acknowledgments
We wish to acknowledge the support of the Engineering and Physical Sciences Research
Council through grants GR/M71404/01 and GR/R97047/01.
Appendix A. Analytical continuation of Bessel functions
Confusion arises with Bessel functions of order 1/3 due to the special care needed to treat
the phase factors: some popular mathematical software uses incorrect continuation formulae.
Firstly our convention is the same as Watson’s so that the appropriate continuation formulae
Field emission theory for an enhanced surface potential
are given by
Iν (z) =
e−iνπ/2 Jν (zeiπ/2 )
e3iνπ/2 Jν (ze−3iπ/2 )
877
if −π < arg(z) π/2
if π/2 < arg(z) π.
An approach to perform the continuation for negative z is
√ √
π x J−1/3 ( 23 x 3/2 )
−2i 3/2
i√ =
|x|
u1 =
π |x|J−1/3
3
3
3
3/2 √
i
2|x|
,
= eiπ/6 π |x|I−1/3
3
3
(A.1)
(A.2)
which is incorrect. The negative argument prevents the use of the first form of equation (A.1).
Instead the branch cut properties require the continuation formula [32]
Jν (eimπ z) = eimνπ Jν (z)
which holds similarly for the Iν functions. Then
√ √
π x J−1/3 ( 23 x 3/2 )
i√ −2i 3/2
=
|x|
π |x|J−1/3
u1 =
3
3
3
3/2
1√
2|x|
=
π |x|I−1/3
,
3
3
(A.3)
(A.4)
without any complex phase factors. Similarly u 2 picks up a minus sign:
√ √
π x J1/3 ( 23 x 3/2 )
i√ −2i 3/2
1√ 2|x|3/2
.(A.5)
=
|x|
=−
π |x|J1/3
π |x|I1/3
u2 =
3
3
3
3
3
Hence one has to manually continue the Jν to the Iν functions for negative arguments when,
if using the incorrect continuation, equation (A.2) results. We have found that Mathematica
v4.1 uses equation (A.2).
Appendix B. Infinite series solutions
The generic form of the full image finite field problem is given by the second-order differential
equation:
(ξ − ¯ )ψ + ξ(ξ − ¯ )ψ + β̄ψ = 0.
(B.1)
Unfortunately this does not belong to the standard hypergeometric class and cannot therefore
be treated by the usual mathematical physics functions. However, using the standard Frobenius
method one can obtain the following power series solutions. Let
u m (ξ ) =
∞
an ξ m+n ,
(B.2)
n=0
then equation (B.2) leads to the following five-term recursion relation for the determination of
the coefficients an :
an−1 (m + n − 1)(m + n − 2) − an ¯ (m + n)(m + n − 1) + an−4 − ¯ an−3 + ᾱan−2 = 0.
(B.3)
All the coefficients are well determined in terms of the coefficient a0 which is a normalization
factor that can be set to 1. The first term satisfies
m(m
¯
− 1)a0 = 0,
(B.4)
878
T C Choy et al
giving m = 0 or 1, which shows, as expected, that we shall have two linearly independent
solutions. Carrying on we have a1 = 0 and
a2 =
a0
β̄
.
¯ (m + 2)(m + 1)
(B.5)
Notice that a1 = 0 regardless but a2 is only finite for β̄ = 0, i.e. it originates from the image
potential. The following coefficients can be sequentially determined:
a0
a2 (m + 1)
−
,
¯ (m + 3)
(m + 3)(m + 2)
a0
β̄
a2
a3 (m + 2)
+
+
,
a4 =
¯ (m + 4) ¯ (m + 4)(m + 3) ¯ (m + 4)(m + 3)
a3 =
(B.6)
(B.7)
and so on. The general five-term recursion relation can be re-written as
1
β̄
1
1
an =
an−1 (m + n − 1)(m + n − 2) + an−2 − an−3 + an−4 . (B.8)
(m + n)(m + n − 1) ¯
¯
¯
Unfortunately there is no way to simplify this any further. When β̄ = 0 (image free case) the
above in fact reduces to a two-term relation given by
an−3
an = −
.
(B.9)
(m + n)(m + n − 1)
This immediately leads us to the two independent solutions, for m = 0:
u 0 (ξ ) = 1 +
∞
(−1)n 1 · 4 · 7 · · · (3n − 2)
(3n)!
n=1
2 3/2
(2/3) 1/2
ξ
,
=
ξ
J
−1/3
31/3
3
ξ 3n
(B.10)
and for m = 1:
u 1 (ξ ) = ξ +
∞
(−1)n 2 · 5 · 8 · · · (3n − 1)
n=1
(3n + 1)!
= 31/3 (4/3)ξ 1/2 J1/3 ( 23 ξ 3/2 ),
ξ 3n+1
(B.11)
which are the series expansions for the Bessel functions of order 1/3. The oscillatory nature of
these functions means that the convergence is, in general, poor and some more sophisticated
re-summation method must be used to obtain, in particular, the asymptotic properties of these
functions. Also an independent solution does not seem to exist for the zero-field finite image
case using the above power series equation (B.2), which is perhaps not surprising, since the
Whittaker functions [2, 18, 40] have a branch point at the origin. However, the re-summation
procedure described below does yield an independent solution, which is unfortunately a
divergent series.
B.1. A method of re-summation
An alternative solution for the full image problem which is an effective re-summation of the
previous power series solution that will allow us to perform numerical calculations, will now
be shown. The approach is to first use a Laplace transform on equation (B.1). Let
eξ t f (t) dt,
(B.12)
ψ(ξ ) =
C
Field emission theory for an enhanced surface potential
879
over some contour C, then the resulting differential equation in ψ becomes converted into a
second-order differential equation in f :
f − (t 2 − ¯ ) f + (β̄ − ¯ t 2 − 2t) f = 0.
(B.13)
t 3 /3
It is easy to show that the solution for β̄ = 0 reduces to the Airy integral where f (t) = e , so
3
we shall factorize f in the form: f = ug, where u = et /3 . The equation for g now simplifies
tremendously:
g + (t 2 + ¯ )g + β̄g = 0.
(B.14)
Unfortunately there is still no closed form solution for this equation although modern methods
of analysis have now yielded some interesting new results. For it can be proved that solutions
for g must be an entire function, by which we can develop an expansion as an infinite series
of the form [38, 39]
∞
g=
an t m+n .
(B.15)
n=0
The boundary condition requires that when β̄ → 0 then g → 1. With this we can show that
m = 0 is the only allowed index. Then we have the following four-term recursion relation:
(n − 1)(nan + ¯ an−1 ) + (n − 3)an−3 + β̄an−2 = 0.
(B.16)
β̄
The first few solutions are easily obtained: a0 = 1, a2 = β̄2 , a3 = ¯6β̄ and a4 = 24
(β̄ − ¯ 2 ) and
so on. As required only the first term survives in the limit β̄ = 0. Interestingly the four-term
n
relation equation (B.16) can actually be reduced to a three-term one. Let γn = aan−1
then
(n − 1)(nγn + ¯ )γn−1 +
(n − 3)
+ β̄ = 0,
γn−2
(B.17)
which may be more convenient for computation. Finally we have now effectively re-summed
the series to obtain
∞
1 3
an
t n eξ t+ 3 t dt.
(B.18)
ψ(ξ ) =
n=0
We now see that
n ξ t+ 13 t 3
t e
C
C
1 3
dn
dt = n
eξ t+ 3 t dt
dξ C
dn
2
= n ξ 1/2 Z ±1/3 ξ 3/2 ,
dξ
3
(B.19)
the last term following from the relation between the Airy functions and the Bessel functions
of order 1/3. In view of the recursion properties for the derivatives of the Bessel functions, we
have now converted the solution effectively into an infinite series of Bessel functions. Only
the first term survives in the image-free case, which is now trivial, and all the oscillatory bits
are now absorbed into the Bessel functions. A further re-summation appears to be necessary
to recover the Whittaker functions in the zero-field finite image β̄ = 0 case.
References
[1]
[2]
[3]
[4]
Fowler R H and Nordheim L W 1928 Proc. R. Soc. A 119 173–81
Nordheim L W 1928 Proc. R. Soc. A 121 626–39
Millikan R A and Lauritsen C C 1928 Proc. Natl Acad. Sci. 14 45
Jeffreys H 1924 Proc. Lond. Math. Soc. 23 428–36
880
[5]
[6]
[7]
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
[18]
[19]
[20]
[21]
[22]
[23]
[24]
[25]
[26]
[27]
[28]
[29]
[30]
[31]
[32]
[33]
[34]
[35]
[36]
[37]
[38]
[39]
[40]
[41]
[42]
[43]
[44]
[45]
[46]
T C Choy et al
Priority for the WKB method appears to go back to
Liouville J 1837 J. Math. 2 16
Liouville J 1837 J. Math. 2 418
and later
Lord Rayleigh 1912 Proc. R. Soc. 86 207
Wentzel G 1926 Z. Phys. 38 518
Kramers H A 1926 Z. Phys. 39 828
Brillouin L 1926 C. R. Acad. Sci. 183 24
Burgess R F, Kroemer H and Houston J M 1953 Phys. Rev. 90 515
Good R H and Müller E W 1956 Handbuch der Physik vol 21, ed S Flugge (Berlin: Springer) pp 176–91
Gadzuk J W and Plummer E W 1973 Rev. Mod. Phys. 45 487
Jensen K L 2001 Theory of field emission Vacuum Microelectronics ed W Zhu (New York: Wiley)
Burden A P 2001 Materials for field emission displays Int. Mater. Rev. 46 213–31
Gao H et al 2003 J. Appl. Phys. 93 5602–5
Hastas N A et al 2002 Semicond. Sci. Technol. 17 662–7
Mott N F and Gurney R W 1948 Electronic Processes in Ionic Crystals (Oxford: Oxford University Press)
Herring C H 1950 Rev. Mod. Phys. 21 187
Dearnaley G, Stoneham A M and Morgan D V 1970 Rep. Prog. Phys. 33 1129
MacColl L A 1939 Phys. Rev. 56 699–702
Sternin B Y and Shatalov V E 1996 Borel–Laplace Transform and Asymptotic Theory—Introduction to Resurgent
Analysis (Boca Raton, FL: CRC Press)
Li Y B, Bando Y and Goldberg D 2003 Appl. Phys. Lett. 82 1962–4
Tuck R A, Taylor W and Latham R V 1997 Proc. 4th Int. Display Workshop (Nagoya: Society for Information
Display) pp 723–6
Guth E and Mullin C J 1940 Phys. Rev. 59 575–84
See, for example, Zangwill A 1992 Physics at Surfaces (Cambridge: Cambridge University Press) pp 57–63
Landau L D and Lifshitz E M 1991 Quantum Mechanics (Oxford: Pergamon) p 165 (section 49)
Jensen K L et al 2002 Appl. Phys. Lett. 81 3867–9
Modinos A 1984 Field, Thermionic and Secondary Emission Spectroscopy (New York: Plenum)
Roy D K 1977 Tunnelling and Negative Resistance Phenomena in Semiconductors (Int. Series on the Science
of the Solid State vol 11) (Oxford: Pergamon) p 53
Delbourgo R 1977 Am. J. Phys. 45 1110–2
Duke C B 1969 Tunnelling in Solids (Solid State Physics Suppl. 10) ed F Seitz, D Turnbull and H Ehrenreich
(New York: Academic)
Moll J L 1964 Physics of Semiconductors (New York: McGraw-Hill)
Landau L D and Lifshitz E M 1991 Quantum Mechanics (Oxford: Pergamon) pp 74–9 (section 24)
Watson G H 1944 Theory of Bessel Functions (Cambridge: Cambridge University Press) p 77
Sze S M 1981 Physics of Semiconductors 2nd edn (New York: Wiley) pp 254–65 and references quoted therein
Murphy E L and Good R H Jr 1956 Phys. Rev. 102 1464–73
Stratton R 1962 Phys. Rev. 125 67–82
Schiff L I 1965 Quantum Mechanics (New York: McGraw-Hill) p 103
Shawyer B and Watson B 1994 Borel’s Methods of Summability (Oxford: Oxford University Press)
Gundersen G G 1988 Trans. Am. Math. Soc. 305 415–29
Belaı̈di B and Hamouda S 2001 Electron. J. Diff. Eqns 2001 1–5
Gradshteyn I S and Ryzhik I M 1980 Table of Integrals, Series and Products (New York: Academic) pp 1059–64
(sections 9.22–23)
Choy T C 2000 Phys. Rev. A 62 12506–16 and references quoted therein
Landau L D, Lifshitz E M and Pitaevskii L P 1984 Electrodynamics of Continuous Media (Oxford: Pergamon)
p 59 (section 15)
Landauer R and Martin Th 1994 Rev. Mod. Phys. 66 217–28
Langhoff P W, Epstein S T and Karplus M 1972 Rev. Mod. Phys. 44 603–42
Choy T C 1999 Effective Medium Theory—Principles and Applications (Oxford: Oxford University Press)
pp 47–71 and references therein
Jaeger D L, Hren J J and Zhirnov V V 2003 J. Appl. Phys. 93 691–7